A copula-based analysis of false discovery rate control under dependence assumptions

ECONOMICS & STATISTICS DISCUSSION PAPER

No. 065/12

A copula-based analysis of false discovery rate control under dependence

assumptions

Roy Cerqueti

Mauro Costantini

Claudio Lupi

The Economics & Statistics Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. The views expressed in the papers are solely the responsibility of the authors.

rate control under dependence assumptions

Abstract

The false discovery rate (FDR) first introduced in Benjamini and Hochberg (1995) is a powerful approach to multiple testing. Benjamini and Yekutieli(2001) proved that the original procedure developed for independent test statistics controls the FDR also for positively dependent test statistics. Furthermore, Yekutieli (2008) showed that a modification of the original procedure can be used even in the presence of non-positively regression dependent test statistics. In this paper we elaborate on Yekutieli (2008) and introduce suitable classes of copulas to identify the conditions under which the dependence properties needed to control the FDR are satisfied.

Keywords: Multiple testing, False discovery rate, Copulas. JEL: C12, C40.

1. Introduction

When many hypotheses are tested simultaneously, the risk of falsely rejecting truly null hy-potheses increases dramatically. In one single test we usually reject the null if the testp-value,

p, is such that p   α, for a pre-specified level α. Since p U_p0,1q under the null, we have

that Prpp   α|H0q α. But when m "1 hypotheses are tested simultaneously it is likely

that at leastone of the p-values is less thanα even if all the hypotheses are truly null. On the other hand, a researcher would probably like to identify as many “discoveries” as possible (Sori´c 1989), while incurring in a small proportion of false positives. This is the motivation of the concept of False Discovery Rate(FDR) introduced by Benjamini and Hochberg (1995). In plain words, the FDR is the expected value of the proportion of errors among the rejected hypotheses.

Benjamini and Yekutieli (2001) proposed a procedure to control the FDR at level q for all joint test statistics, under a positive dependence. The proposed strategy consists in the application of the Benjamini-Hochberg (BH) (Benjamini and Hochberg 1995) scheme at level

qL °m_i1i1. An even more general procedure, the separate subsets BH (ssBH) procedure, has been introduced recently in Yekutieli(2008) in order to deal with more general forms of dependence.

In order to fix the ideas, we offer here a brief explanation of how the ssBH procedure works. Denote by p pp1, . . . , pmq1 the vector of the m p values associated with the components of

the collection of mstatistics t pt1, . . . , tmq1. Consistently withYekutieli (2008), we assume

that the p values in p are co-monotone transformations of the corresponding test statistics in t. Divide p in S sub-vectors ps_{, for s 1, . . . , S. With a very intuitive notation, the}

statistics corresponding topsconstitute a vector, that will be indicated withts. Assume that the cardinality ofps isms and denote asps₀ thep values inpscorresponding to the true null hypotheses. The levelq ssBH procedure runs into two steps as follows:

1. Fors1, . . . , S, apply the BH procedure at levelqms_{{m to testp}s_{, and denote asr}s BH

the pvalues corresponding to the rejected hypotheses.

2. Reject the null hypothesis corresponding to rssBH

”S

s1rBHs .

This paper aims at developing formal arguments to characterise the relationship between FDR control and the dependence properties of the individual statistics involved, when the ssBH procedure advocated in Yekutieli (2008) is used. To achieve our aim, we model the stochastic dependence among the univariate statistics through the introduction of suitable families of copulas. In doing so, we are able to deal with dependence concepts more general than Pearson’s correlation or those based on linearity, which are classically related to the limited world of normal random variables.

The rest of the paper is organized as follows. Section 2 concisely provides the statistical background. The main results are contained in Section 3. A discussion of the meaning and worthiness of the results is offered in Section4. The proofs are collected in Section5.

2. Statistical preliminaries

We stress that FDR control is strongly related to the stochastic dependence among the in-dividual test statistics belonging to t. Therefore, a detailed discussion on the dependence structure underlying thet’s is needed. In this respect and to be self-contained, we recall here the concept ofpositive regression dependency on each one from a subset I0 „ t1, . . . , mu or,

briefly,PRDS on I0:

Definition 2.1. Consider an increasing set1 D. The vectortis assumed to satisfy the PRDS onI0 if, for each iPI0, the conditional probability PrptPD|ti xq is nondecreasing in x.

Benjamini and Yekutieli(2001) proved that the PRDS property on subsets of the test statistics

t’s corresponding to the true null hypothesis ensures the control of the FDR at a certain level by the BH procedure. Unfortunately, this result meets severe drawbacks in practice, because of the difficulty in showing the PRDS property. To overcome this problem, themultivariate total positivity of order 2 or, briefly, MTP2 property — a stronger dependence structure — can be used instead. We recall here the definition of MTP2:

Definition 2.2. Let f be the joint density function of the m-variate random variable t. tis said to be MTP2 if and only if, for each x andy in Rm, it results:

fpxq fpyq ¥fpmintx,yuq fpmaxtx,yuq

where the min and max operators have to be intended componentwise.

A well known statistical result ensures that MTP2 ùñ PRDS on I0 @I0. Therefore, the

dependence described by the MTP2 can be used instead of the PRDS onI0, having in mind

that the former condition is stronger.

Furthermore, the (linear) dependence between the individual t’s in t can also be well rep-resented by a non-diagonal variance-covariance matrix pσi,jqi,j1,...,m, where variances are

indicated with a unique index as: σi,i σi2. Hence, it is natural to guess a relationship

between the value of the covariances and the MTP2 dependence property. In this respect, it is useful to recall a further definition of dependence between random variables.

Definition 2.3. The random variables tt1, . . . , tmu are associated if

Covpgpt1, . . . , tmq, hpt1, . . . , tmqq ¥0,

for any coordinatewise nondecreasing functionsg, h:RmÑRfor which this covariance exists.

A classical result states that random variables that are MTP2 are also associated. Therefore, by using Definition2.3withgpt1, . . . , tmq ti andhpt1, . . . , tmq tj, the relationship between

the MTP2 property and covariance states immediately as follows:

Proposition 2.4. Assume that tt1, . . . , tmu are MTP2. Then σi,j ¥ 0, for each i, j

1, . . . , m.

Note that Proposition 2.4 implies that if there exists a couple such that σi,j   0, then

tt1, . . . , tmu are not MTP2.

A rather general way to capture the stochastic dependence structure among random variables is the introduction of the concept of multivariate copula(or, simply, copula). We now provide the definition of this concept, referring the reader toNelsen (2006) for a detailed discussion.

Definition 2.5. The function C:r0,1snÑ r0,1s is a copula if and only if: (C2.5.i)Cpu1, . . . , unq 0 if u1. . .un0;

(C2.5.ii)Cpu1, . . . , unq u¯_k if uk 1, for each k¯k;

(C2.5.iii) Given then-dimensional rectangle ra1, b1s . . . ran, bns „ r0,1sn, then 2 ¸ i11 . . . 2 ¸ in1 p1qi1 ... in_Cp_u 1,i1, . . . , un,inq ¥0, where uj,1aj anduj,2 bj.

The classical Sklar’s Theorem (Sklar 1959) highlights how multivariate copulas introduced in Definition2.5model the dependence structure between random variables. For the sake of completeness, we report here the enunciation of the theorem, adapted to our case:

Theorem 2.6(sklar1959). LetFi1,...,is be the joint distribution function of thes-plepti1, . . . , tisq,

with i1, . . . , is 1, . . . , m. Define the margins as Fi1, . . . , Fis. Then there exists a s-variate

copula Ci1,...,is such that, for eachx1, . . . , xs PR,

Fi1,...,ispx1, . . . , xsq Ci1,...,ispFi1px1q, . . . , Fispxsqq. (1)

If the margins Fi1, . . . , Fis are continuous, then the copula Ci1,...,is is unique. Conversely,

if Ci1,...,is is a s-variate copula and Fi1, . . . , Fis are distribution functions, then the function

Theorem2.6points out that, given a set of random variables, the relationship between joint and marginal distributions is stated through copulas.

In the presence of bivariate copulas, i.e. for s 2, we can derive the covariance between couples of random variables:

Proposition 2.7. σi,j 1 σiσj » » R2 rCi,jpFipxq, Fjpyqq FipxqFjpyqsdxdy,

where Ci,j is the bivariate copula defined as in (1).

Proposition 2.7 shows that the covariance between couples of random variables pti, tjq can

be derived from the knowledge of the copula describing their stochastic dependence. More generally, we can say that the introduction of a multivariate copula leads to the identification of a variance-covariance matrix.

3. Main results

The argument on the stochastic dependence developed above can be applied to the FDR control of the multiple statisticst. In order to proceed, we need a condition on the sets ps

introduced above:

Condition 3.1. One of the following assumptions holds:

(A3.1.i) if pi Pp0, then there exists a uniquesi P t1, . . . , Su such thatpiPpsi. Moreover, for

eachs1, . . . , S, it must be:

ms

"

2, ifps₀ H; arbitrary, otherwise.

(A3.1.ii) psiX_psj H_{, for s}

i sj, and ms2, for each s1, . . . , S.

Condition3.1means that the division of the setpin the subsets ps is such that each pvalue of a true null hypothesis is contained in one ps, and each ps containing a p value of a true null hypothesis has cardinality equals to 2. This is not a restrictive hypothesis, since the decomposition oftpsus1,...,S to be used for the ssBH procedure can be arbitrarily chosen. It

is worth noting that when (A3.1.ii) holds, then ms 2, for each s 1, . . . , S; if (A3.1.i) is true, thenDS˜¤S such thatms_{2, for eachs1, . . . ,S}˜_.

We are now able to state our first main result:

Proposition 3.2. Assume that Condition 3.1 holds and that the dependence between the statistics in ts is described by a copula Cs such that:

Cspu, vq uv θφpuqφpvq, (2)

for eachs1, . . . , S, withθP r1,1sandφ:r0,1s Ñ r0,1ssatisfying the following conditions: (C3.2.i)φp0q φp1q 0;

(C3.2.ii)φ is Lipschitzian in r0,1s, i.e.: |φpxq φpyq| ¤ |xy|, for eachx, yP r0,1s; (C3.2.iii)φ is convex or concave inr0,1s.

The main limitations of this approach are basically two. First, Proposition 3.2 refers to couples, i.e. subsets of cardinality 2; furthermore, copula in (2) is symmetric with respect to its arguments, i.e.: Cspu, vq Cspv, uq for each u, v. The latter aspect is a very strong

requirement for multivariate modelling, in that symmetric copulas are able to cover only a small range of dependencies.

However, it is possible to generalize the result by using as-variate approach, with s¡2, in a not necessarily symmetric framework. A copula approach will be adopted also in this case to ensure FDR control.

We first extend the analysis to cover the multivariate case, with s ¡2. Then we provide a generalization to the asymmetric setting.

To deal with thes-variate symmetric framework, it is useful to recall two definitions:

Definition 3.3. Consider a continuous strictly decreasing convex function

ψ:r0,1s Ñ r0, 8q (3)

such that ψp1q 0 and lim

xÑ0 ψpxq 8.

Ans-variate Archimedean copula with generator ψ is a copula Cspψq such that

C_sψpu1, . . . , usq ψ1 _s ¸ i1 ψpuiq . (4)

Analogously to what already noted for copula in (2), also copula Csψ in (4) refers to a

sym-metric case.

We now introduce a generalization of the monotonic property for functions:

Definition 3.4. A function

ψ:r0,1s Ñ r0, 8q (5)

is completely monotone in r0,1s if and only if ψ PC8p0,1q XC0r0,1s, and p1qnψpnqpxq ¥

0,@n0,1,2, . . .; @x P p0,1q.

The following result states a sufficient condition for FDR control in thes-variate symmetric case:

Proposition 3.5. Assume that the dependence between the statistics in ts is described by an Archimedean copulaCsψ, i.e.:

"

Csψpu1, . . . , usq ψ1p

°s

i1ψpuiqq;

ukFkpxkq, xkPR, @k1, . . . , s. , (6)

where ψ is completely monotone in [0,1].

Then the level q BH procedure controls the FDR at levelqm0{m.

The generalization to the asymmetric framework can be obtained at the cost of some mildly stronger assumptions. We enter the details, by firstly introducing an asymmetric copula constituting a generalization of the Archimedean copula proposed in (4).

Definition 3.6. Let us introduce a set of sm functions

hjk :r0,1s Ñ r0,1s, j1, . . . , m; k1, . . . , s (7)

such that:

(C3.6.i)hjk is differentiable and strictly increasing, for each j, k;

(C3.6.ii)hjkp0q 0 and hjkp1q 1;

(C3.6.iii) _m1 °m_j1hjkpxq x, for each k1, . . . , s andxP r0,1s.

Moreover, define

ψ:r0,1s Ñ r0,1s (8)

such that:

(C3.6.iv) ψ iss 2 times differentiable inp0,1q; (C3.6.v) ψpiq¡0, for i1, . . . , s;

(C3.6.vi) ψp0q 0 and ψp1q 1. We define an (Archimedean) asymmetric copula as C_ASψ :

r0,1ss Ñ r0,1ssuch that: C_ASψ pu1, . . . , usq ψ1 1 m m ¸ j1 s ¹ k1 hjkpψpukqq . (9)

Copula in (9) has been firstly introduced and explored in Liebscher (2008). It is worth noting that, as far as the copula’s definition is concerned, conditions (C3.6.iv) and (C3.6.v) could be weakened, our stronger version being required to prove the following general result:

Proposition 3.7. Assume that

ψ1ps 2qpxq ψ1psqpxq ψ1ps 1qpxq 2 ¥0, @xP p0,1q (10) and ψpukq hjk1 euk ₁ e1 , j 1, . . . , m; k1, . . . , s. (11) Moreover, suppose that the dependence between the statistics in ts _{is described by copula (9).}

Then the level q BH procedure controls the FDR at levelq0{m.

4. Discussion

The introduction of copulas to model stochastic dependence allows us to deal with more general dependence structures than Pearson’s correlation or those based on linearity, which are appropriate only when referring to normal multivariate models. In this respect, it is worth noting that covariance can be derived directly from copulas (see Proposition 2.7), but the converse is not true.

In our framework, Propositions3.2,3.5, and3.7providesufficient conditions for the FDR to hold in the presence of fairly general dependence schemes.

Proposition 3.2 offers viable ways of selecting the couples in such a way that the conditions for validly using the ssBH procedure are satisfied. The main limitations are related to the

cardinality of the subsets and to the condition of symmetry between couples of random vari-ables. As a good feature, we must notice that the pairwise dependence introduced in the set up given by family oftpsus1,...,S, Condition 3.1 and the copulas in (2) allow us to describe

a system with both positively and negatively correlated test statistics. Indeed, the positive dependence condition formalized by TP2 (see Proposition2.4) is required only for some pairs of statistics in t (the ones appearing in the ts’s), while no assumptions are stated on the remaining couples. This aspect meets a natural requirement on the dependence structure of statistics in multiple testing.

Furthermore, copula defined in (2) allows to derive an explicit expression for the correlation between the individual statistics in ts. Indeed, some algebra provides that if the stochastic dependence between X and Y is described through copula Cs in (2), then the correlation

coefficientρX,Y betweenX and Y can be written as:

ρX,Y 12θ »1 0 φpξqdξ 2 . (12)

Notice also that copula (2) used in Proposition3.2is a “perturbation” of the product copula: when θ 0 the case collapses to independence. It is also worth noting that copula in (2) is a generalization of the Farlie-Gumbel-Morgestern (FGM) copula that holds when φpuq up1uq. In the bivariate case, when θ¥0, it implies positive quadrant dependence (see Lai and Xie 2000), which is a weaker form of dependence than TP2. However, a word of caution is in order here. The FGM copula, as well as its studied variants, are known for implying only modest dependence (see, e.g.,Huang and Kotz 1999): therefore, we cannot expect the copula (2) to accurately represent very strong dependence across the test statistics. As far as the “pure” FGM copula is concerned, its dependence as measured by Kendall’s τ and Spearman’s

ρis respectively 2θ{9 andθ{3 with1¤θ¤1. However, an appropriate choice of a function

φpuq up1uqsatisfying conditions in Proposition3.2allows to strengthen the typical weak dependence structure of the FGM copula.

The need to obtain a truly multivariate result motivates the formulation of Proposition 3.5. As with the bivariate case, it is worth noting that the introduction of the family of sets

tpsus1,...,S with the stochastic dependence structure formalized in Proposition3.5 allows us

to describe a system with both positively and negatively correlated test statistics. Indeed, the positive dependence condition formalized by MTP2 is required only for some statistics in t

(the ones appearing in thets’s), while no assumptions are stated on the remaining statistics. Furthermore, the Archimedean copulas (6) used in Proposition3.5are more flexible than the particular case of FGM copulas in that they can represent cases with both strong positive and negative dependence. Kendall’s τ for Archimedean copulas takes the convenient form (seeGenest and MacKay 1986, Theorem 2)

τX,Y 4

»1

0

ψpξq

ψ1pξqdξ 1. (13)

Copulas in (2) and in (6) exhibit a symmetry property, in that they are invariant with respect to permutation of their univariate arguments. Such a symmetry is able to model dependence structures which depend only on a small number of parameters. This is their main limitation, in that they are not particularly flexible in fitting multivariate data with a large number of parameters. It is also worth noting that symmetric copulas are able to model only a rather small range of dependencies.

For these reasons, the extension to the asymmetric case has been proposed. Proposition

3.7 extends the FDR applicability to situations where dependence can be well represented by asymmetric copulas (see Liebscher 2008, 2011, on asymmetric copulas). In this respect, Proposition3.7 complements and extendsYekutieli(2008).

5. Proofs

In this section we provide the proofs for our main results.

Proposition 3.2

Proof. Conditions (C3.2.i) and (C3.2.ii) guarantee that Cs in (2) is a copula.

Denote as X and Y the individual statistics in ts. Amblard and Girard (2002) shows that, if the dependence betweenX and Y is described through the copulaCs in (2) and condition

(C3.2.iii) holds, thenY is stochastically increasing in X andX is stochastically increasing in

Y, i.e. the following conditions hold:

"

PpY ¡y|X xq is nondecreasing inx, @y;

PpX¡x|Y yq is nondecreasing iny, @x. (14)

The system (14) is equivalent to the TP2 property for the set ts _{(see Nelsen 2006). Hence,}

Condition3.1and Proposition 2.2 inYekutieli (2008) give the thesis.

Proposition 3.5

Proof. Denote as t1, . . . , ts the individual statistics in ts. If the stochastic dependence in ts

is described as in the hypotheses, thenM¨uller and Scarsini(2005) guarantees that the MTP2 property holds fort1, . . . ,s. Hence, Proposition 2.2 in Yekutieli(2008) gives the thesis.

Proposition 3.7

Proof. We need to prove that C_ASψ satisfies the MTP2 property. Then, Proposition 2.2 in

Yekutieli(2008) gives the thesis.

In virtue of M¨uller and Scarsini (2005), it is sufficient to check that the density f of C_ASψ is log-supermodular, that is equivalent to say that

logpfpu1, . . . , usqq:log Bs Bu1. . .Bus C_ASψ pu1, . . . , usq (15) is supermodular.

By (9) we have fpu1, . . . , usq B s Bu1. . .Bus C_ASψ pu1, . . . , usq ψr1s psq 1 m m ¸ j1 s ¹ k1 hjkpψpukqq 1 m m ¸ j1 s ¹ k1 h1_jkpψpukqqψ1pukq. (16) By (16) we can write logpfpu1, . . . , usqq log ψr1s psq 1 m m ¸ j1 s ¹ k1 hjkpψpukqq log 1 m m ¸ j1 s ¹ k1 h1_jkpψpukqqψ1pukq : Apu1, . . . , usq Bpu1, . . . , usq, (17)

where the termsApqand Bpq are an intuitive shorthand for the two logrsterms. The supermodularity of logrfpu1, . . . , usqsis equivalent to the following condition:

B2

Buk1Buk2

rApu1, . . . , usq Bpu1, . . . , usqs ¥0, (18)

for each k1, k2 P t1, . . . , su, and pu1, . . . , usq P r0,1ss. For an easier notation, we will pose

hereafter ξ: 1 m m ¸ j1 s ¹ k1 hjkpψpukqq. (19)

We analyse the termsApq and Bpq separately. First notice that

BApu1, . . . , usq Buk1 ψr1s ps 1q pξq ψr1spsq_pξq 1 m m ¸ j1 h1_jk1pψpuk1qqψ1puk1q ¹ kk1 rhjkpψpukqqs

and B2_Apu 1, . . . , usq Buk1Buk2 # 1 m m ¸ j1 h1_jk1pψpuk1qqψ 1_pu k1q ¹ kk1 rhjkpψpukqqs + # 1 m m ¸ j1 h1_jk1pψpuk2qqψ1puk2q ¹ kk2 rhjkpψpukqqs + #ψr1s ps 2qpξq ψr1s psqpξq ψr1s ps 1qpξq 2+ ψr1s psqpξq ψr1s ps 1qpξq 1 m m ¸ j1 h1_jk1pψpuk1qqψ 1_pu k1qh 1 jk2pψpuk2qqψ 1_pu k2q ¹ kk1,k2 rhjkpψpukqqs . (20)

Hence, under condition (C3.6.v) and hypothesis (10), we have

B2_Apu

1, . . . , usq

Buk1Buk2

¥0. (21)

Let us now turn toBpq:

BBpu1, . . . , usq Buk1 °m j1 h2_jk 1pψpuk1qqψ1puk1q h1jk1pψpuk1qqψ2puk1q ± kk1h 1 jkpψpukqqψ1pukq °m j1 ±s k1h1jkpψpukqqψ1pukq , hence we have: B2_Bpu 1, . . . , usq Buk1Buk2 °m j1 ± kk1,k2h 1 jkpψpukqqψ1pukq °m j1 ±s k1h1jkpψpukqqψ1pukq 2 ! ¹ kk1,k2 rh2_jkpψpukqqpψ1pukqq2 h1jkpψpukqqψ2pukqs 1 m m ¸ j1 s ¹ k1 h1_jkpψpukqqψ1pukq h2_jk1pψpuk1qqpψ 1_pu k1qq 2 _h1 jk1pψpuk1qqψ 2_pu k1q h1_jk2pψpuk2qqψ1puk2q 1 m m ¸ j1 h2_jk2pψpuk2qqpψ1puk2qq 2 _h1 jk2pψpuk2qqψ2puk2q ¹ kk2 h1_jkpψpukqqψ1pukq ) . (22)

By (22) we obtain that a sufficient condition for beingB2Bpu1, . . . , usq{ pBuk1Buk2q 0 is: h2_jk2pψpuk2qqpψ1puk2qq

2 _h1

We then need to solve the ODE in (23), with the initial condition given by (C3.6.vi). Note that (23) is equivalent to

h2_jk 2pψpuk2qqψ1puk2q h1_jk 2pψpuk2qq ψ2puk2q ψ1puk2q ψ1puk2q ,

which leads to:

logrh1_jk2pψpuk2qqs » ψ2puk2q ψ1puk2q ψ1puk2q duk2. Then h1_jk2pψpuk2qq H1euk2 ψ1puk2q , H1 P p0, 8q. (24)

(24) can be rewritten as follows:

h1_jk2pψpuk2qq ψ

1_pu

k2q H1e

uk2,

which immediately leads to

hjk2pψpuk2qq H1e

uk2 H₂, H₂ P_R.

Sincehjk2 is invertible, by imposing the initial conditions in (C3.6.vi) we obtain:

ψpuk2q h 1 jk2 euk_{2 1} e1 ,

that is exactly condition (11) when kk2.

The result is proved, by the arbitrariness of k2.

Acknowledgements

We thank Rachele Foschi and Fabio Spizzichino for comments and discussion. Materials related to this paper were presented at the Conference in honour of Professor M.H. Pesaran (Cambridge, July 2011): we are grateful to conference participants for their constructive comments.

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Affiliation:

Roy Cerqueti

University of Macerata, Dept. of Economic and Financial Institutions.

Via Crescimbeni, 20. I-62100 Macerata, Italy E-mail: [email protected]

Mauro Costantini

Brunel University, Dept. of Economics and Finance.

Kingston Lane, Uxbridge. Middlesex UB8 3PH, United Kingdom E-mail: [email protected]

Claudio Lupi

University of Molise, Dept. of Economics, Management, and Social Sciences. Via De Sanctis. I-86100 Campobasso, Italy